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slide 2 Nonlinear damping is important in MEMS Motivation: Micro plates are very important in many microsystems applications. Squeezed-film damping determines the dynamics of plates moving a few microns above the substrate. Examples abound in MEMS accelerometers. MEMS switches. MEMS gyroscopes. Measurement of damping in MEMS has not been as extensively explored as the modeling. Published measurements have not addressed the nonlinearity inherent to the variation of the thickness of the squeezed film gap throughout the oscillation cycle. Methods for measuring nonlinear dynamic responses are not sensitive enough to measure damping nonlinearity. Objective: Provide experimental method to measure time-varying damping on MEMS.

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slide 4 The oscillating plate can be modeled as SDOF. Equation of motion Squeeze-film damping was theoretically predicted to be nonlinear. Free vibration of the plate resembles a decaying sinusoid. Higher when plate is closer to substrate. Lower when plate is farther from substrate. Textbook log-decrement method cannot capture nonlinear damping.

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slide 6 Hilbert transform gives decay envelope. The Hilbert transform of a signal v(t) is is an imaginary signal that is 90 o lagging from phase from the real signal v(t). The |vector sum| of the real signal and the imaginary signal is the amplitude The amplitude of a decaying sinusoid is the envelope v(t)v(t) |V|(t)

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slide 14 Nonlinear damping was orders of magnitude lower than linear damping. Amplitude as a Function of Time: Fit Values Versus Measured Data. Total Damping Ratio as a Function of Time. Linear fit matched the envelope very closely.

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slide 15 Nonlinear damping was captured as a function of displacement and velocity. Nonlinear Damping Ratio as a Function of Displacement. Nonlinear Damping Ratio as a Function of Velocity. The method extracted the nonlinear part of the damping ratio, even though that part was orders of magnitude smaller than the linear part. The nonlinear damping appears to be a function of velocity rather than displacement.

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slide 16 Conclusions: The measurement and data processing technique resulted in accurate estimates of oscillation frequency and linear damping ratios. Linear damping ratio can be obtained by curve-fitting the decay envelope. The method extracted the nonlinear part of the damping ratio, even though that part was orders of magnitude smaller than the linear part. The nonlinear damping appears to be a function of velocity rather than displacement.

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slide 17 Acknowledgment Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. The authors thank the following contributors: Chris Dyck and Bill Cowan’s team for providing the test structures. Dan Rader for technical guidance and programmatic support. hSumali@Sandia.gov

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slide 19 Differentiation can result in large noise. Then the oscillation frequency can be obtained as the analytical derivative of that representation. Differentiation with time resulted in unacceptably large noise. Curve fitting of the signal phase can be done to obtain a very close representation of the phase angle as an analytical function of time.